Hello, doctorgk!

A rectangle made of elastic material will be made into a cylinder by joining edges AD and BC.

To support the structure, a wire of fixed length L is placed along the diagonal of the rectangle.

Find the angle $\displaystyle \theta\;(\angle CAB)$ that will maximize the volume of the cylinder. Code:

D * - - - - - - - * C
| * |
| * |
| L * |
| * | L·sinθ
| * |
| * |
| * θ |
A * - - - - - - - * B
L·cosθ

We have: .$\displaystyle AB \,=\,L\cos\theta,\;CB \,=\,L\sin\theta$

The height of the cylinder is: .$\displaystyle h \:=\:CB \:=\:L\sin\theta\;\;{\color{blue}[1]}$

$\displaystyle AB$ is the circumference of the circular base of the cylinder.

. . $\displaystyle L\cos\theta \,=\,2\pi r\quad\Rightarrow\quad r \:=\:\frac{L\cos\theta}{2\pi}\;\;{\color{blue}[2]}$

The volume of a cylinder is: .$\displaystyle V \;=\;\pi r^2h\;\;{\color{blue}[3]}$

Substitute [1] and [2] into [3]: .$\displaystyle V \;=\;\pi\left(\frac{L\cos\theta}{2\pi}\right)^2(L\ sin\theta) $

. . and we have: . $\displaystyle \boxed{V \;=\;\frac{L^3}{4\pi}\,\sin\theta\cos^2\!\theta}$

And **tha**t is the function to be maximized . . . good luck!